Calculus Examples

$\frac{- 8 x^{4} + 3 x^{2} - 9 x}{x}$

Step 1

Simplify terms.

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Step 1.1

Factor $x$ out of $- 8 x^{4} + 3 x^{2} - 9 x$ .

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Step 1.1.1

Factor $x$ out of $- 8 x^{4}$ .

$\frac{d}{d x} [\frac{x (- 8 x^{3}) + 3 x^{2} - 9 x}{x}]$

Step 1.1.2

Factor $x$ out of $3 x^{2}$ .

$\frac{d}{d x} [\frac{x (- 8 x^{3}) + x (3 x) - 9 x}{x}]$

Step 1.1.3

Factor $x$ out of $- 9 x$ .

$\frac{d}{d x} [\frac{x (- 8 x^{3}) + x (3 x) + x \cdot - 9}{x}]$

Step 1.1.4

Factor $x$ out of $x (- 8 x^{3}) + x (3 x)$ .

$\frac{d}{d x} [\frac{x (- 8 x^{3} + 3 x) + x \cdot - 9}{x}]$

Step 1.1.5

Factor $x$ out of $x (- 8 x^{3} + 3 x) + x \cdot - 9$ .

$\frac{d}{d x} [\frac{x (- 8 x^{3} + 3 x - 9)}{x}]$

Step 1.2

Cancel the common factor of $x$ .

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Step 1.2.1

Cancel the common factor.

$\frac{d}{d x} [\frac{x (- 8 x^{3} + 3 x - 9)}{x}]$

Step 1.2.2

Divide $- 8 x^{3} + 3 x - 9$ by $1$ .

$\frac{d}{d x} [- 8 x^{3} + 3 x - 9]$

Step 2

By the Sum Rule, the derivative of $- 8 x^{3} + 3 x - 9$ with respect to $x$ is $\frac{d}{d x} [- 8 x^{3}] + \frac{d}{d x} [3 x] + \frac{d}{d x} [- 9]$ .

$\frac{d}{d x} [- 8 x^{3}] + \frac{d}{d x} [3 x] + \frac{d}{d x} [- 9]$

Step 3

Since $- 8$ is constant with respect to $x$ , the derivative of $- 8 x^{3}$ with respect to $x$ is $- 8 \frac{d}{d x} [x^{3}]$ .

$- 8 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [3 x] + \frac{d}{d x} [- 9]$

Step 4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$- 8 (3 x^{2}) + \frac{d}{d x} [3 x] + \frac{d}{d x} [- 9]$

Step 5

Multiply $3$ by $- 8$ .

$- 24 x^{2} + \frac{d}{d x} [3 x] + \frac{d}{d x} [- 9]$

Step 6

Since $3$ is constant with respect to $x$ , the derivative of $3 x$ with respect to $x$ is $3 \frac{d}{d x} [x]$ .

$- 24 x^{2} + 3 \frac{d}{d x} [x] + \frac{d}{d x} [- 9]$

Step 7

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$- 24 x^{2} + 3 \cdot 1 + \frac{d}{d x} [- 9]$

Step 8

Multiply $3$ by $1$ .

$- 24 x^{2} + 3 + \frac{d}{d x} [- 9]$

Step 9

Since $- 9$ is constant with respect to $x$ , the derivative of $- 9$ with respect to $x$ is $0$ .

$- 24 x^{2} + 3 + 0$

Step 10

Add $- 24 x^{2} + 3$ and $0$ .

$- 24 x^{2} + 3$